Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$, then the value of $\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }$ is

• A. $\frac{p(3q-p^2)}{q}$
• B. $\frac{q(3p-q^2)}{p}$
• C. $\frac{p(3q+p^2)}{q}$
• D. None of these

Answer 1

The correct option is A $\frac{p(3q-p^2)}{q}$

Since $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$
$\therefore \alpha +\beta =-p$ and $\alpha \beta =q$
Now, $\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }=\frac{{\alpha }^3+{\beta }^3}{\alpha \beta }=\frac{(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )}{\alpha \beta }=\frac{-p^3-3q(-p)}{q}=\frac{p(3q-p^2)}{q}$.